ACM Certificate issued for an private hosted zone, status stuck on pending validation

Hi, we have a certificate issued by ACM for the domain for renewal, and the status of this certificate is pending validation. I tried to add the CNAME record by Create Records in Route 53, and it pop as "the record is successfully created". Waited for a day, the certificate is still on pending validation status.

To give a clear example, naming the domain that needs verification as api.example.com. I checked in route53 that there is no CNAME record in the hosted zone "api.example.com", however we have a record in the hosted zone example.com. I'm not sure about the relationship of this two domain names. But api.example.com is a **private hosted zone ** and example.com is a **public hosted zone ** that has the CNAME record we need to add to api.example.com. The record in the public hosted zone has CNAME has record name.

I have followed DNS validation and Why is my AWS Certificate Manager (ACM) certificate DNS validation status still pending validation? and it's a bit confusing that is this the correct certificate we get for a private hosted zone? Shouldn't we get it from ACM CA? If no, where should I add this record to?

I pasted the example below, how do I complete validation for this domain?